Implicit Density Estimation by Local Moment Matching to Sample from Auto-Encoders

Recent work suggests that some auto-encoder variants do a good job of capturing the local manifold structure of the unknown data generating density. This paper contributes to the mathematical understanding of this phenomenon and helps define better justified sampling algorithms for deep learning based on auto-encoder variants. We consider an MCMC where each step samples from a Gaussian whose mean and covariance matrix depend on the previous state, defines through its asymptotic distribution a target density. First, we show that good choices (in the sense of consistency) for these mean and covariance functions are the local expected value and local covariance under that target density. Then we show that an auto-encoder with a contractive penalty captures estimators of these local moments in its reconstruction function and its Jacobian. A contribution of this work is thus a novel alternative to maximum-likelihood density estimation, which we call local moment matching. It also justifies a recently proposed sampling algorithm for the Contractive Auto-Encoder and extends it to the Denoising Auto-Encoder

Local Component Analysis

Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectation-minimization (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semi-parametric Gaussian-Parzen density model, where some of the variables are modelled through a jointly Gaussian density, while others are modelled through Parzen windows. For these two models, EM leads to simple closed-form updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher test-likelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on such metrics, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a large-scale setting

A Graph-Based Semi-Supervised k Nearest-Neighbor Method for Nonlinear Manifold Distributed Data Classification

$k$ Nearest Neighbors ($k$NN) is one of the most widely used supervised learning algorithms to classify Gaussian distributed data, but it does not achieve good results when it is applied to nonlinear manifold distributed data, especially when a very limited amount of labeled samples are available. In this paper, we propose a new graph-based $k$NN algorithm which can effectively handle both Gaussian distributed data and nonlinear manifold distributed data. To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by constructing an $R$-level nearest-neighbor strengthened tree over the graph, and then compute a TRW matrix for similarity measurement purposes. After this, the nearest neighbors are identified according to the TRW matrix and the class label of a query point is determined by the sum of all the TRW weights of its nearest neighbors. To deal with online situations, we also propose a new algorithm to handle sequential samples based a local neighborhood reconstruction. Comparison experiments are conducted on both synthetic data sets and real-world data sets to demonstrate the validity of the proposed new $k$NN algorithm and its improvements to other version of $k$NN algorithms. Given the widespread appearance of manifold structures in real-world problems and the popularity of the traditional $k$NN algorithm, the proposed manifold version $k$NN shows promising potential for classifying manifold-distributed data.Comment: 32 pages, 12 figures, 7 table

Novel Perspectives and Applications of Knowledge Graph Embeddings: From Link Prediction to Risk Assessment and Explainability

Knowledge graph representation is an important embedding technology that supports a variety of machine learning related applications. By learning the distributed representation of multi-relational data, knowledge embedding models are supposed to efficiently deal with the semantic relatedness of their constituents. However, failing in the fundamental task of creating an appropriate form to represent knowledge harms any attempt of designing subsequent machine learning tasks. Several knowledge embedding methods have been proposed in the last decade. Although there is a consensus on the idea that enhanced approaches are more efficient, more complex projections in the hyperspace that indeed favor link prediction (or knowledge graph completion) can result in a loss of semantic similarity. We propose a new evaluation task that aims at performing risk assessment on domain-specific categorized multi-relational datasets, designed as a classification problem based on the resulting embeddings. We assess the quality of embedding representations based on the synergy of the resulting clusters of target subjects. We show that more sophisticated embedding approaches do not necessarily favor embedding quality, and the traditional link prediction validation protocol is a weak metric to measure the quality of embedding representation. Finally, we present insights about using the synergy analysis to provide risk assessment explainability based on the probability distribution of feature-value pairs within embedded clusters

Local Component Analysis

International audienceKernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectation-minimization (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semi-parametric Gaussian-Parzen density model, where some of the variables are modelled through a jointly Gaussian density, while others are modelled through Parzen windows. For these two models, EM leads to simple closed-form updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher test-likelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on such metrics, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a large-scale setting